Subgroups — C₂×C₆₇×#2155 #30712654

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2155	C₂	1	8	yes	maximal, direct product factor
5	C₂×#365×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₆₇	—	2	1	no	—
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₇	—	2	1	no	—
9	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₆₇	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
14	C₂×#108×C₆₇	—	2	1	no	—
15	C₂×#108×C₆₇	—	2	2	no	—
16	C₂×#108×C₆₇	C₂²	1	2	yes	—
17	C₂×#108×C₆₇	C₂²	1	4	yes	—
18	#427×C₆₇	—	2	2	no	—
19	#427×C₆₇	C₂²	1	4	yes	—
20	C₂×#107×C₆₇	—	2	1	no	—
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	#365×C₆₇	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	4	no	—
26	C₂×M₁₆×C₆₇	—	4	1	no	—
27	C₂×M₁₆×C₆₇	—	4	2	no	—
28	C₂²×C₈×C₆₇	—	4	1	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	2	yes	—
34	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
36	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
39	#117×C₆₇	—	2	4	no	—
40	#117×C₆₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
43	C₂×C₄²×C₆₇	—	4	1	no	—
44	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
45	M₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
46	M₃₂×C₆₇	C₂³	1	4	yes	—
47	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
49	#108×C₆₇	—	2	2	no	—
50	#108×C₆₇	—	2	4	no	—
51	#108×C₆₇	C₂³	1	4	yes	—
52	#107×C₆₇	—	2	4	no	—
53	#106×C₆₇	—	2	2	no	—
54	#106×C₆₇	C₂³	1	4	yes	—
55	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	M₁₆×C₆₇	—	2	4	no	—
68	M₁₆×C₆₇	—	4	2	no	—
69	M₁₆×C₆₇	—	4	4	no	—
70	C₂×C₈×C₆₇	—	4	2	no	—
71	C₂×C₈×C₆₇	—	4	4	no	—
72	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	4	2	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	—	4	4	no	—
94	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#106	1	1	yes	characteristic
97	C₂³×C₆₇	#106	1	1	yes	characteristic
98	C₂³×C₆₇	#106	1	2	yes	—
99	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	8	4	no	—
103	C₂×C₄×C₆₇	#106	1	2	yes	—
104	C₂×C₄×C₆₇	#106	1	2	yes	—
105	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#106	1	2	yes	—
107	C₂×C₄×C₆₇	#106	1	2	yes	—
108	C₂×C₄×C₆₇	#106	1	2	yes	—
109	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₆₇	—	4	4	no	—
122	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	C₂×#106	1	2	yes	—
134	C₂²×C₆₇	C₂×#106	1	2	yes	—
135	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
137	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₆₇	—	8	4	no	—
139	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#106	1	2	yes	—
141	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#106	1	2	yes	—
143	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂×#2155	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	#2155	1	2	yes	direct product factor
151	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
152	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic
153	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
154	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
155	#2155	C₂×C₆₇	1	8	yes	direct product factor
156	C₂×#365	C₂×C₆₇	1	4	yes	complemented
157	C₆₇	C₂×#2155	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×M₁₆	—	2	1	no	—
159	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×#117	—	2	1	no	—
161	C₂×M₃₂	C₄×C₆₇	1	2	yes	complemented
162	C₂×M₃₂	C₄×C₆₇	1	4	yes	complemented
163	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
164	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
165	C₂²×C₁₆	C₄×C₆₇	1	2	yes	complemented
166	C₂×#108	—	2	1	no	—
167	C₂×#108	—	2	2	no	—
168	C₂×#108	C₂²×C₆₇	1	2	yes	—
169	C₂×#108	C₂²×C₆₇	1	4	yes	—
170	#427	—	2	2	no	—
171	#427	C₂²×C₆₇	1	4	yes	—
172	C₂×#107	—	2	1	no	—
173	C₂×#106	C₂²×C₆₇	1	2	yes	—
174	#365	C₂²×C₆₇	1	16	yes	complemented
175	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	4	1	no	—
179	C₂×M₁₆	—	4	2	no	—
180	C₂²×C₈	—	4	1	no	—
181	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
184	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
185	C₂²×C₈	D₄×C₆₇	1	2	yes	—
186	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
188	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
191	#117	—	2	4	no	—
192	#117	—	4	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	4	1	no	—
195	C₂×C₄²	—	4	1	no	—
196	M₃₂	C₂×C₄×C₆₇	1	4	yes	—
197	M₃₂	C₂×C₄×C₆₇	1	8	yes	complemented
198	M₃₂	C₂³×C₆₇	1	4	yes	—
199	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
200	C₂×C₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
201	#108	—	2	2	no	—
202	#108	—	2	4	no	—
203	#108	C₂³×C₆₇	1	4	yes	—
204	#107	—	2	4	no	—
205	#106	—	2	2	no	—
206	#106	C₂³×C₆₇	1	4	yes	—
207	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
211	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
214	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
226	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
227	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
228	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
229	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
230	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
237	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
239	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
240	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
241	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₁₆	C₂²×C₄×C₆₇	1	8	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#106×C₆₇	1	1	yes	characteristic
249	C₂³	#106×C₆₇	1	1	yes	characteristic
250	C₂³	#106×C₆₇	1	2	yes	—
251	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
252	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	#106×C₆₇	1	2	yes	—
256	C₂×C₄	#106×C₆₇	1	2	yes	—
257	C₂×C₄	#106×C₆₇	1	2	yes	—
258	C₂×C₄	#106×C₆₇	1	2	yes	—
259	C₂×C₄	#106×C₆₇	1	2	yes	—
260	C₂×C₄	#106×C₆₇	1	1	yes	characteristic
261	C₂×C₄	#106×C₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
267	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
270	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
276	C₈	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
277	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
278	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
279	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#106×C₆₇	1	2	yes	—
286	C₂²	C₂×#106×C₆₇	1	2	yes	—
287	C₂²	C₂×#106×C₆₇	1	1	yes	characteristic
288	C₂²	C₂×#106×C₆₇	1	1	yes	characteristic
289	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	C₂×#106×C₆₇	1	1	yes	characteristic
292	C₄	C₂×#106×C₆₇	1	2	yes	—
293	C₄	C₂×#106×C₆₇	1	1	yes	characteristic
294	C₄	C₂×#106×C₆₇	1	2	yes	—
295	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
296	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₆₇×#2155	1	2	yes	direct product factor
302	C₂	C₂²×#106×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#106
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂⁴×C₆₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₆₇×#2155	C₂	direct (N × H)
5. C₂×#365×C₆₇	C₂	semidirect
9. C₂×M₃₂×C₆₇	C₄	semidirect
10. C₂×M₃₂×C₆₇	C₄	semidirect
13. C₂²×C₁₆×C₆₇	C₄	semidirect
22. #365×C₆₇	C₂²	semidirect
45. M₃₂×C₆₇	C₂×C₄	semidirect
48. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
145. C₂×#2155	C₆₇	direct (N × H)
150. C₂×C₆₇	#2155	direct (N × H)
155. #2155	C₂×C₆₇	direct (N × H)
156. C₂×#365	C₂×C₆₇	semidirect
157. C₆₇	C₂×#2155	direct (N × H)
161. C₂×M₃₂	C₄×C₆₇	semidirect
162. C₂×M₃₂	C₄×C₆₇	semidirect
165. C₂²×C₁₆	C₄×C₆₇	semidirect
174. #365	C₂²×C₆₇	semidirect
197. M₃₂	C₂×C₄×C₆₇	semidirect
200. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
301. C₂	C₆₇×#2155	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×M₃₂×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#427×C₆₇	2
C₆₇×#2155	2
C₂×#365×C₆₇	2
C₂×#2155	67

Subgroups — C2×C67×#2155 #30712654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2155 #30712654